We present the results of new ROSAT observations of four quasars. Two radio-loud objects, 3C110 and 3C334, and the radio-quiet quasar, Mkn 877, have relatively soft X-ray spectra (alpha(X) almost-equal-to 1.2), whereas a third radio-loud object, PKS0403-132, has a canonical X-ray continuum slope (alpha(X) almost-equal-to 0.7). The latter source has the most core-dominated radio structure of the sample consistent with the previously established correlation of X-ray spectral hardness with radio core-dominance in quasars, which has been interpreted in terms of the relativistic beaming of a relatively hard X-ray spectral component. However, ROSAT observations in the 0.1 - 2 keV band are generally indicating much softer X-ray spectra than were measured previously with the Einstein observatory in the 0.3 - 3.5 keV band for all classes of quasar including those with core-dominated radio structure. We argue that this is quite difficult to explain in terms of models which envisage a mixture of beamed and unbeamed X-ray components. We also report new measurements of the galactic neutral hydrogen column density in the direction of 49 radio-loud quasars.